My mother gave me a book called “Making Mathematics with Needlework” last Christmas, and I fell in love with the idea of making a Sierpinski fractal shawl. Here’s a nice picture and explanation of the Sierpinski triangle”. Basically you take a triangle, bisect the sides to cut out a smaller triangle, and then you have three triangles; keep doing it. My shawl has 5 iterations, and I crocheted it from the bottom tip up. It’s not equilateral! But it still works.

The basic pattern is this: each row increases by 1 arch. The row starts and finishes with 2 arches, which are made by chaining 5 stitches and then attaching it to the arch (or filled fan) below it by a single-crochet. The filled pieces (fans) are made by triple-crocheting thrice into the same stitch and then attaching in a similar way. I don’t like this pattern all that much because it ultimately doesn’t make triangles so much as curvy diamonds, but it worked well overall. It took me forever, though, because each row just kept getting longer! I used a variegated baby yarn of sport weight and a size G hook.

An overall view. It's about 3 feet from base to tip, but the base stretches to about 5 feet.

A closer up view. It's upside-down from the way I actually crocheted it, though.

so you always add the two numbers in one row to get the number in the next row, except that you put a 0 (a hole) whenever the number is even and a 1 (filled space) when it’s odd. That way you get
1
1 1
1 0 1
1 1 1 1
1 0 0 0 1
1 1 0 0 1 1
1 0 1 0 1 0 1
1 1 1 1 1 1 1 1

and so on. So you can build it from the ground up instead of cutting pieces out. Whee, math geekiness!